How do I clean the interior lining of a leather purse that cannot be turned inside out?

I bought a leather purse secondhand, and there are some stains on the inside. Unfortunately, the lining is stitched down on all sides, so I can’t just pull the lining out. Most of the instructions and video tutorials I’ve found online only show linings that stitched at the top and can be pulled outward. I haven’t been able to find one like mine.

I could probably clean the main compartment by turning the entire bag inside out, but there are a bunch of flatter side pockets that would have to accessed “right side out.” My main concern is over saturating and staining/ruining the leather exterior.

Any suggestions or links to video/written instructions are much appreciated!